Editing Polyomino (section)

====Inductive algorithms====
Each polyomino of size ''n''+1 can be obtained by adding a square to a polyomino of size ''n''. This leads to algorithms for generating polyominoes inductively.

Most simply, given a list of polyominoes of size ''n'', squares may be added next to each polyomino in each possible position, and the resulting polyomino of size ''n''+1 added to the list if not a duplicate of one already found; refinements in ordering the enumeration and marking adjacent squares that should not be considered reduce the number of cases that need to be checked for duplicates.<ref>Golomb, pp. 73–79</ref> This method may be used to enumerate either free or fixed polyominoes.

A more sophisticated method, described by Redelmeier, has been used by many authors as a way of not only counting polyominoes (without requiring that all polyominoes of size ''n'' be stored in size to enumerate those of size ''n''+1), but also proving upper bounds on their number. The basic idea is that we begin with a single square, and from there, recursively add squares. Depending on the details, it may count each ''n''-omino ''n'' times, once from starting from each of its ''n'' squares, or may be arranged to count each once only.

The simplest implementation involves adding one square at a time. Beginning with an initial square, number the adjacent squares, clockwise from the top, 1, 2, 3, and 4. Now pick a number between 1 and 4, and add a square at that location. Number the unnumbered adjacent squares, starting with 5. Then, pick a number larger than the previously picked number, and add that square. Continue picking a number larger than the number of the current square, adding that square, and then numbering the new adjacent squares. When ''n'' squares have been created, an ''n''-omino has been created.

This method ensures that each fixed polyomino is counted exactly ''n'' times, once for each starting square. It can be optimized so that it counts each polyomino only once, rather than ''n'' times. Starting with the initial square, declare it to be the lower-left square of the polyomino. Simply do not number any square that is on a lower row, or left of the square on the same row. This is the version described by Redelmeier.

If one wishes to count free polyominoes instead, then one may check for symmetries after creating each ''n''-omino. However, it is faster<ref>Redelmeier, section 4</ref> to generate symmetric polyominoes separately (by a variation of this method)<ref>Redelmeier, section 6</ref> and so determine the number of free polyominoes by [[Burnside's lemma]].